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Within the framework of free actions of compact quantum groups on unital C*-algebras, we propose two conjectures. The first one states that, if $\delta :A\to A{⨂}_{min}H$ is a free coaction of the C*-algebra H of a non-trivial compact quantum group on a unital C*-algebra A, then there is no H-equivariant *-homomorphism from A to the equivariant join C*-algebra $A{⊛}_{\delta }H$. For A being the C*-algebra of continuous functions on a sphere with the antipodal coaction of the C*-algebra of functions on ℤ/2ℤ, we recover the celebrated Borsuk-Ulam...